One generator quasi-cyclic codes over F2 + uF2

Abstract

In this paper, we study quasi-cyclic codes over the ring R = F-2 + uF(2) = {0, 1, u, u + 1} where u(2) = 0. By exploring their structure, we determine the type of one generator quasi-cyclic codes over R and the size by giving a minimal spanning set. We also determine the rank and introduce a lower bound for the minimum distance of free quasi-cyclic codes over R. We include some examples of quasi-cyclic codes of various lengths over R. In particular, we obtain a family of 2-quasi-cyclic codes from cyclic codes over the ring F-2 + uF(2) + vF(2) + uvF(2). Finally, using the Gray map we obtain a family of optimal binary linear codes as the images of quasi-cyclic codes over R. (C) 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.